Left Termination of the query pattern perm_in_2(g, a) w.r.t. the given Prolog program could successfully be proven:

0 Prolog
  ↳1 PrologToPiTRSProof (⇐)
    ↳2 PiTRS
    ↳3 DependencyPairsProof (⇔)
    ↳4 PiDP
    ↳5 DependencyGraphProof (⇔)
    ↳6 AND
      ↳7 PiDP
        ↳8 UsableRulesProof (⇔)
        ↳9 PiDP
        ↳10 PiDPToQDPProof (⇐)
        ↳11 QDP
        ↳12 QDPSizeChangeProof (⇔)
        ↳13 TRUE
      ↳14 PiDP
        ↳15 UsableRulesProof (⇔)
        ↳16 PiDP
        ↳17 PiDPToQDPProof (⇐)
        ↳18 QDP
        ↳19 QDPSizeChangeProof (⇔)
        ↳20 TRUE
      ↳21 PiDP
        ↳22 UsableRulesProof (⇔)
        ↳23 PiDP
        ↳24 PiDPToQDPProof (⇐)
        ↳25 QDP
        ↳26 MRRProof (⇔)
        ↳27 QDP
        ↳28 PisEmptyProof (⇔)
        ↳29 TRUE
  ↳30 PrologToPiTRSProof (⇐)
    ↳31 PiTRS
    ↳32 DependencyPairsProof (⇔)
    ↳33 PiDP
    ↳34 DependencyGraphProof (⇔)
    ↳35 AND
      ↳36 PiDP
        ↳37 UsableRulesProof (⇔)
        ↳38 PiDP
        ↳39 PiDPToQDPProof (⇐)
        ↳40 QDP
        ↳41 QDPSizeChangeProof (⇔)
        ↳42 TRUE
      ↳43 PiDP
        ↳44 UsableRulesProof (⇔)
        ↳45 PiDP
        ↳46 PiDPToQDPProof (⇐)
        ↳47 QDP
        ↳48 QDPSizeChangeProof (⇔)
        ↳49 TRUE
      ↳50 PiDP
        ↳51 UsableRulesProof (⇔)
        ↳52 PiDP
        ↳53 PiDPToQDPProof (⇐)
        ↳54 QDP

(0) Obligation:

Clauses:

app1(.(X, Xs), Ys, .(X, Zs)) :- app1(Xs, Ys, Zs).
app1([], Ys, Ys).
app2(.(X, Xs), Ys, .(X, Zs)) :- app2(Xs, Ys, Zs).
app2([], Ys, Ys).
perm(Xs, .(X, Ys)) :- ','(app2(X1s, .(X, X2s), Xs), ','(app1(X1s, X2s, Zs), perm(Zs, Ys))).
perm([], []).

Queries:

perm(g,a).

(1) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
perm_in: (b,f)
app2_in: (f,f,b)
app1_in: (b,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

perm_in_ga(Xs, .(X, Ys)) → U3_ga(Xs, X, Ys, app2_in_aag(X1s, .(X, X2s), Xs))
app2_in_aag(.(X, Xs), Ys, .(X, Zs)) → U2_aag(X, Xs, Ys, Zs, app2_in_aag(Xs, Ys, Zs))
app2_in_aag([], Ys, Ys) → app2_out_aag([], Ys, Ys)
U2_aag(X, Xs, Ys, Zs, app2_out_aag(Xs, Ys, Zs)) → app2_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_ga(Xs, X, Ys, app2_out_aag(X1s, .(X, X2s), Xs)) → U4_ga(Xs, X, Ys, X1s, X2s, app1_in_gga(X1s, X2s, Zs))
app1_in_gga(.(X, Xs), Ys, .(X, Zs)) → U1_gga(X, Xs, Ys, Zs, app1_in_gga(Xs, Ys, Zs))
app1_in_gga([], Ys, Ys) → app1_out_gga([], Ys, Ys)
U1_gga(X, Xs, Ys, Zs, app1_out_gga(Xs, Ys, Zs)) → app1_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(Xs, X, Ys, X1s, X2s, app1_out_gga(X1s, X2s, Zs)) → U5_ga(Xs, X, Ys, perm_in_ga(Zs, Ys))
perm_in_ga([], []) → perm_out_ga([], [])
U5_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) → perm_out_ga(Xs, .(X, Ys))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
app2_in_aag(x1, x2, x3)  =  app2_in_aag(x3)
.(x1, x2)  =  .(x2)
U2_aag(x1, x2, x3, x4, x5)  =  U2_aag(x5)
app2_out_aag(x1, x2, x3)  =  app2_out_aag(x1, x2)
U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x6)
app1_in_gga(x1, x2, x3)  =  app1_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x5)
[]  =  []
app1_out_gga(x1, x2, x3)  =  app1_out_gga(x3)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x4)
perm_out_ga(x1, x2)  =  perm_out_ga(x2)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(2) Obligation:

Pi-finite rewrite system:
The TRS R consists of the following rules:

perm_in_ga(Xs, .(X, Ys)) → U3_ga(Xs, X, Ys, app2_in_aag(X1s, .(X, X2s), Xs))
app2_in_aag(.(X, Xs), Ys, .(X, Zs)) → U2_aag(X, Xs, Ys, Zs, app2_in_aag(Xs, Ys, Zs))
app2_in_aag([], Ys, Ys) → app2_out_aag([], Ys, Ys)
U2_aag(X, Xs, Ys, Zs, app2_out_aag(Xs, Ys, Zs)) → app2_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_ga(Xs, X, Ys, app2_out_aag(X1s, .(X, X2s), Xs)) → U4_ga(Xs, X, Ys, X1s, X2s, app1_in_gga(X1s, X2s, Zs))
app1_in_gga(.(X, Xs), Ys, .(X, Zs)) → U1_gga(X, Xs, Ys, Zs, app1_in_gga(Xs, Ys, Zs))
app1_in_gga([], Ys, Ys) → app1_out_gga([], Ys, Ys)
U1_gga(X, Xs, Ys, Zs, app1_out_gga(Xs, Ys, Zs)) → app1_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(Xs, X, Ys, X1s, X2s, app1_out_gga(X1s, X2s, Zs)) → U5_ga(Xs, X, Ys, perm_in_ga(Zs, Ys))
perm_in_ga([], []) → perm_out_ga([], [])
U5_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) → perm_out_ga(Xs, .(X, Ys))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
app2_in_aag(x1, x2, x3)  =  app2_in_aag(x3)
.(x1, x2)  =  .(x2)
U2_aag(x1, x2, x3, x4, x5)  =  U2_aag(x5)
app2_out_aag(x1, x2, x3)  =  app2_out_aag(x1, x2)
U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x6)
app1_in_gga(x1, x2, x3)  =  app1_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x5)
[]  =  []
app1_out_gga(x1, x2, x3)  =  app1_out_gga(x3)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x4)
perm_out_ga(x1, x2)  =  perm_out_ga(x2)

(3) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

PERM_IN_GA(Xs, .(X, Ys)) → U3_GA(Xs, X, Ys, app2_in_aag(X1s, .(X, X2s), Xs))
PERM_IN_GA(Xs, .(X, Ys)) → APP2_IN_AAG(X1s, .(X, X2s), Xs)
APP2_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → U2_AAG(X, Xs, Ys, Zs, app2_in_aag(Xs, Ys, Zs))
APP2_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → APP2_IN_AAG(Xs, Ys, Zs)
U3_GA(Xs, X, Ys, app2_out_aag(X1s, .(X, X2s), Xs)) → U4_GA(Xs, X, Ys, X1s, X2s, app1_in_gga(X1s, X2s, Zs))
U3_GA(Xs, X, Ys, app2_out_aag(X1s, .(X, X2s), Xs)) → APP1_IN_GGA(X1s, X2s, Zs)
APP1_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → U1_GGA(X, Xs, Ys, Zs, app1_in_gga(Xs, Ys, Zs))
APP1_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APP1_IN_GGA(Xs, Ys, Zs)
U4_GA(Xs, X, Ys, X1s, X2s, app1_out_gga(X1s, X2s, Zs)) → U5_GA(Xs, X, Ys, perm_in_ga(Zs, Ys))
U4_GA(Xs, X, Ys, X1s, X2s, app1_out_gga(X1s, X2s, Zs)) → PERM_IN_GA(Zs, Ys)

The TRS R consists of the following rules:

perm_in_ga(Xs, .(X, Ys)) → U3_ga(Xs, X, Ys, app2_in_aag(X1s, .(X, X2s), Xs))
app2_in_aag(.(X, Xs), Ys, .(X, Zs)) → U2_aag(X, Xs, Ys, Zs, app2_in_aag(Xs, Ys, Zs))
app2_in_aag([], Ys, Ys) → app2_out_aag([], Ys, Ys)
U2_aag(X, Xs, Ys, Zs, app2_out_aag(Xs, Ys, Zs)) → app2_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_ga(Xs, X, Ys, app2_out_aag(X1s, .(X, X2s), Xs)) → U4_ga(Xs, X, Ys, X1s, X2s, app1_in_gga(X1s, X2s, Zs))
app1_in_gga(.(X, Xs), Ys, .(X, Zs)) → U1_gga(X, Xs, Ys, Zs, app1_in_gga(Xs, Ys, Zs))
app1_in_gga([], Ys, Ys) → app1_out_gga([], Ys, Ys)
U1_gga(X, Xs, Ys, Zs, app1_out_gga(Xs, Ys, Zs)) → app1_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(Xs, X, Ys, X1s, X2s, app1_out_gga(X1s, X2s, Zs)) → U5_ga(Xs, X, Ys, perm_in_ga(Zs, Ys))
perm_in_ga([], []) → perm_out_ga([], [])
U5_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) → perm_out_ga(Xs, .(X, Ys))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
app2_in_aag(x1, x2, x3)  =  app2_in_aag(x3)
.(x1, x2)  =  .(x2)
U2_aag(x1, x2, x3, x4, x5)  =  U2_aag(x5)
app2_out_aag(x1, x2, x3)  =  app2_out_aag(x1, x2)
U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x6)
app1_in_gga(x1, x2, x3)  =  app1_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x5)
[]  =  []
app1_out_gga(x1, x2, x3)  =  app1_out_gga(x3)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x4)
perm_out_ga(x1, x2)  =  perm_out_ga(x2)
PERM_IN_GA(x1, x2)  =  PERM_IN_GA(x1)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x4)
APP2_IN_AAG(x1, x2, x3)  =  APP2_IN_AAG(x3)
U2_AAG(x1, x2, x3, x4, x5)  =  U2_AAG(x5)
U4_GA(x1, x2, x3, x4, x5, x6)  =  U4_GA(x6)
APP1_IN_GGA(x1, x2, x3)  =  APP1_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5)  =  U1_GGA(x5)
U5_GA(x1, x2, x3, x4)  =  U5_GA(x4)

We have to consider all (P,R,Pi)-chains

(4) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

PERM_IN_GA(Xs, .(X, Ys)) → U3_GA(Xs, X, Ys, app2_in_aag(X1s, .(X, X2s), Xs))
PERM_IN_GA(Xs, .(X, Ys)) → APP2_IN_AAG(X1s, .(X, X2s), Xs)
APP2_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → U2_AAG(X, Xs, Ys, Zs, app2_in_aag(Xs, Ys, Zs))
APP2_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → APP2_IN_AAG(Xs, Ys, Zs)
U3_GA(Xs, X, Ys, app2_out_aag(X1s, .(X, X2s), Xs)) → U4_GA(Xs, X, Ys, X1s, X2s, app1_in_gga(X1s, X2s, Zs))
U3_GA(Xs, X, Ys, app2_out_aag(X1s, .(X, X2s), Xs)) → APP1_IN_GGA(X1s, X2s, Zs)
APP1_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → U1_GGA(X, Xs, Ys, Zs, app1_in_gga(Xs, Ys, Zs))
APP1_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APP1_IN_GGA(Xs, Ys, Zs)
U4_GA(Xs, X, Ys, X1s, X2s, app1_out_gga(X1s, X2s, Zs)) → U5_GA(Xs, X, Ys, perm_in_ga(Zs, Ys))
U4_GA(Xs, X, Ys, X1s, X2s, app1_out_gga(X1s, X2s, Zs)) → PERM_IN_GA(Zs, Ys)

The TRS R consists of the following rules:

perm_in_ga(Xs, .(X, Ys)) → U3_ga(Xs, X, Ys, app2_in_aag(X1s, .(X, X2s), Xs))
app2_in_aag(.(X, Xs), Ys, .(X, Zs)) → U2_aag(X, Xs, Ys, Zs, app2_in_aag(Xs, Ys, Zs))
app2_in_aag([], Ys, Ys) → app2_out_aag([], Ys, Ys)
U2_aag(X, Xs, Ys, Zs, app2_out_aag(Xs, Ys, Zs)) → app2_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_ga(Xs, X, Ys, app2_out_aag(X1s, .(X, X2s), Xs)) → U4_ga(Xs, X, Ys, X1s, X2s, app1_in_gga(X1s, X2s, Zs))
app1_in_gga(.(X, Xs), Ys, .(X, Zs)) → U1_gga(X, Xs, Ys, Zs, app1_in_gga(Xs, Ys, Zs))
app1_in_gga([], Ys, Ys) → app1_out_gga([], Ys, Ys)
U1_gga(X, Xs, Ys, Zs, app1_out_gga(Xs, Ys, Zs)) → app1_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(Xs, X, Ys, X1s, X2s, app1_out_gga(X1s, X2s, Zs)) → U5_ga(Xs, X, Ys, perm_in_ga(Zs, Ys))
perm_in_ga([], []) → perm_out_ga([], [])
U5_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) → perm_out_ga(Xs, .(X, Ys))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
app2_in_aag(x1, x2, x3)  =  app2_in_aag(x3)
.(x1, x2)  =  .(x2)
U2_aag(x1, x2, x3, x4, x5)  =  U2_aag(x5)
app2_out_aag(x1, x2, x3)  =  app2_out_aag(x1, x2)
U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x6)
app1_in_gga(x1, x2, x3)  =  app1_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x5)
[]  =  []
app1_out_gga(x1, x2, x3)  =  app1_out_gga(x3)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x4)
perm_out_ga(x1, x2)  =  perm_out_ga(x2)
PERM_IN_GA(x1, x2)  =  PERM_IN_GA(x1)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x4)
APP2_IN_AAG(x1, x2, x3)  =  APP2_IN_AAG(x3)
U2_AAG(x1, x2, x3, x4, x5)  =  U2_AAG(x5)
U4_GA(x1, x2, x3, x4, x5, x6)  =  U4_GA(x6)
APP1_IN_GGA(x1, x2, x3)  =  APP1_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5)  =  U1_GGA(x5)
U5_GA(x1, x2, x3, x4)  =  U5_GA(x4)

We have to consider all (P,R,Pi)-chains

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 5 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

APP1_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APP1_IN_GGA(Xs, Ys, Zs)

The TRS R consists of the following rules:

perm_in_ga(Xs, .(X, Ys)) → U3_ga(Xs, X, Ys, app2_in_aag(X1s, .(X, X2s), Xs))
app2_in_aag(.(X, Xs), Ys, .(X, Zs)) → U2_aag(X, Xs, Ys, Zs, app2_in_aag(Xs, Ys, Zs))
app2_in_aag([], Ys, Ys) → app2_out_aag([], Ys, Ys)
U2_aag(X, Xs, Ys, Zs, app2_out_aag(Xs, Ys, Zs)) → app2_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_ga(Xs, X, Ys, app2_out_aag(X1s, .(X, X2s), Xs)) → U4_ga(Xs, X, Ys, X1s, X2s, app1_in_gga(X1s, X2s, Zs))
app1_in_gga(.(X, Xs), Ys, .(X, Zs)) → U1_gga(X, Xs, Ys, Zs, app1_in_gga(Xs, Ys, Zs))
app1_in_gga([], Ys, Ys) → app1_out_gga([], Ys, Ys)
U1_gga(X, Xs, Ys, Zs, app1_out_gga(Xs, Ys, Zs)) → app1_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(Xs, X, Ys, X1s, X2s, app1_out_gga(X1s, X2s, Zs)) → U5_ga(Xs, X, Ys, perm_in_ga(Zs, Ys))
perm_in_ga([], []) → perm_out_ga([], [])
U5_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) → perm_out_ga(Xs, .(X, Ys))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
app2_in_aag(x1, x2, x3)  =  app2_in_aag(x3)
.(x1, x2)  =  .(x2)
U2_aag(x1, x2, x3, x4, x5)  =  U2_aag(x5)
app2_out_aag(x1, x2, x3)  =  app2_out_aag(x1, x2)
U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x6)
app1_in_gga(x1, x2, x3)  =  app1_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x5)
[]  =  []
app1_out_gga(x1, x2, x3)  =  app1_out_gga(x3)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x4)
perm_out_ga(x1, x2)  =  perm_out_ga(x2)
APP1_IN_GGA(x1, x2, x3)  =  APP1_IN_GGA(x1, x2)

We have to consider all (P,R,Pi)-chains

(8) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(9) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

APP1_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APP1_IN_GGA(Xs, Ys, Zs)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
APP1_IN_GGA(x1, x2, x3)  =  APP1_IN_GGA(x1, x2)

We have to consider all (P,R,Pi)-chains

(10) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(11) Obligation:

Q DP problem:
The TRS P consists of the following rules:

APP1_IN_GGA(.(Xs), Ys) → APP1_IN_GGA(Xs, Ys)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(12) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

  • APP1_IN_GGA(.(Xs), Ys) → APP1_IN_GGA(Xs, Ys)
    The graph contains the following edges 1 > 1, 2 >= 2

(13) TRUE

(14) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

APP2_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → APP2_IN_AAG(Xs, Ys, Zs)

The TRS R consists of the following rules:

perm_in_ga(Xs, .(X, Ys)) → U3_ga(Xs, X, Ys, app2_in_aag(X1s, .(X, X2s), Xs))
app2_in_aag(.(X, Xs), Ys, .(X, Zs)) → U2_aag(X, Xs, Ys, Zs, app2_in_aag(Xs, Ys, Zs))
app2_in_aag([], Ys, Ys) → app2_out_aag([], Ys, Ys)
U2_aag(X, Xs, Ys, Zs, app2_out_aag(Xs, Ys, Zs)) → app2_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_ga(Xs, X, Ys, app2_out_aag(X1s, .(X, X2s), Xs)) → U4_ga(Xs, X, Ys, X1s, X2s, app1_in_gga(X1s, X2s, Zs))
app1_in_gga(.(X, Xs), Ys, .(X, Zs)) → U1_gga(X, Xs, Ys, Zs, app1_in_gga(Xs, Ys, Zs))
app1_in_gga([], Ys, Ys) → app1_out_gga([], Ys, Ys)
U1_gga(X, Xs, Ys, Zs, app1_out_gga(Xs, Ys, Zs)) → app1_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(Xs, X, Ys, X1s, X2s, app1_out_gga(X1s, X2s, Zs)) → U5_ga(Xs, X, Ys, perm_in_ga(Zs, Ys))
perm_in_ga([], []) → perm_out_ga([], [])
U5_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) → perm_out_ga(Xs, .(X, Ys))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
app2_in_aag(x1, x2, x3)  =  app2_in_aag(x3)
.(x1, x2)  =  .(x2)
U2_aag(x1, x2, x3, x4, x5)  =  U2_aag(x5)
app2_out_aag(x1, x2, x3)  =  app2_out_aag(x1, x2)
U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x6)
app1_in_gga(x1, x2, x3)  =  app1_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x5)
[]  =  []
app1_out_gga(x1, x2, x3)  =  app1_out_gga(x3)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x4)
perm_out_ga(x1, x2)  =  perm_out_ga(x2)
APP2_IN_AAG(x1, x2, x3)  =  APP2_IN_AAG(x3)

We have to consider all (P,R,Pi)-chains

(15) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(16) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

APP2_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → APP2_IN_AAG(Xs, Ys, Zs)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
APP2_IN_AAG(x1, x2, x3)  =  APP2_IN_AAG(x3)

We have to consider all (P,R,Pi)-chains

(17) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(18) Obligation:

Q DP problem:
The TRS P consists of the following rules:

APP2_IN_AAG(.(Zs)) → APP2_IN_AAG(Zs)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(19) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

  • APP2_IN_AAG(.(Zs)) → APP2_IN_AAG(Zs)
    The graph contains the following edges 1 > 1

(20) TRUE

(21) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

U3_GA(Xs, X, Ys, app2_out_aag(X1s, .(X, X2s), Xs)) → U4_GA(Xs, X, Ys, X1s, X2s, app1_in_gga(X1s, X2s, Zs))
U4_GA(Xs, X, Ys, X1s, X2s, app1_out_gga(X1s, X2s, Zs)) → PERM_IN_GA(Zs, Ys)
PERM_IN_GA(Xs, .(X, Ys)) → U3_GA(Xs, X, Ys, app2_in_aag(X1s, .(X, X2s), Xs))

The TRS R consists of the following rules:

perm_in_ga(Xs, .(X, Ys)) → U3_ga(Xs, X, Ys, app2_in_aag(X1s, .(X, X2s), Xs))
app2_in_aag(.(X, Xs), Ys, .(X, Zs)) → U2_aag(X, Xs, Ys, Zs, app2_in_aag(Xs, Ys, Zs))
app2_in_aag([], Ys, Ys) → app2_out_aag([], Ys, Ys)
U2_aag(X, Xs, Ys, Zs, app2_out_aag(Xs, Ys, Zs)) → app2_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_ga(Xs, X, Ys, app2_out_aag(X1s, .(X, X2s), Xs)) → U4_ga(Xs, X, Ys, X1s, X2s, app1_in_gga(X1s, X2s, Zs))
app1_in_gga(.(X, Xs), Ys, .(X, Zs)) → U1_gga(X, Xs, Ys, Zs, app1_in_gga(Xs, Ys, Zs))
app1_in_gga([], Ys, Ys) → app1_out_gga([], Ys, Ys)
U1_gga(X, Xs, Ys, Zs, app1_out_gga(Xs, Ys, Zs)) → app1_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(Xs, X, Ys, X1s, X2s, app1_out_gga(X1s, X2s, Zs)) → U5_ga(Xs, X, Ys, perm_in_ga(Zs, Ys))
perm_in_ga([], []) → perm_out_ga([], [])
U5_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) → perm_out_ga(Xs, .(X, Ys))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
app2_in_aag(x1, x2, x3)  =  app2_in_aag(x3)
.(x1, x2)  =  .(x2)
U2_aag(x1, x2, x3, x4, x5)  =  U2_aag(x5)
app2_out_aag(x1, x2, x3)  =  app2_out_aag(x1, x2)
U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x6)
app1_in_gga(x1, x2, x3)  =  app1_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x5)
[]  =  []
app1_out_gga(x1, x2, x3)  =  app1_out_gga(x3)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x4)
perm_out_ga(x1, x2)  =  perm_out_ga(x2)
PERM_IN_GA(x1, x2)  =  PERM_IN_GA(x1)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x4)
U4_GA(x1, x2, x3, x4, x5, x6)  =  U4_GA(x6)

We have to consider all (P,R,Pi)-chains

(22) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(23) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

U3_GA(Xs, X, Ys, app2_out_aag(X1s, .(X, X2s), Xs)) → U4_GA(Xs, X, Ys, X1s, X2s, app1_in_gga(X1s, X2s, Zs))
U4_GA(Xs, X, Ys, X1s, X2s, app1_out_gga(X1s, X2s, Zs)) → PERM_IN_GA(Zs, Ys)
PERM_IN_GA(Xs, .(X, Ys)) → U3_GA(Xs, X, Ys, app2_in_aag(X1s, .(X, X2s), Xs))

The TRS R consists of the following rules:

app1_in_gga(.(X, Xs), Ys, .(X, Zs)) → U1_gga(X, Xs, Ys, Zs, app1_in_gga(Xs, Ys, Zs))
app1_in_gga([], Ys, Ys) → app1_out_gga([], Ys, Ys)
app2_in_aag(.(X, Xs), Ys, .(X, Zs)) → U2_aag(X, Xs, Ys, Zs, app2_in_aag(Xs, Ys, Zs))
app2_in_aag([], Ys, Ys) → app2_out_aag([], Ys, Ys)
U1_gga(X, Xs, Ys, Zs, app1_out_gga(Xs, Ys, Zs)) → app1_out_gga(.(X, Xs), Ys, .(X, Zs))
U2_aag(X, Xs, Ys, Zs, app2_out_aag(Xs, Ys, Zs)) → app2_out_aag(.(X, Xs), Ys, .(X, Zs))

The argument filtering Pi contains the following mapping:
app2_in_aag(x1, x2, x3)  =  app2_in_aag(x3)
.(x1, x2)  =  .(x2)
U2_aag(x1, x2, x3, x4, x5)  =  U2_aag(x5)
app2_out_aag(x1, x2, x3)  =  app2_out_aag(x1, x2)
app1_in_gga(x1, x2, x3)  =  app1_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x5)
[]  =  []
app1_out_gga(x1, x2, x3)  =  app1_out_gga(x3)
PERM_IN_GA(x1, x2)  =  PERM_IN_GA(x1)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x4)
U4_GA(x1, x2, x3, x4, x5, x6)  =  U4_GA(x6)

We have to consider all (P,R,Pi)-chains

(24) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(25) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U3_GA(app2_out_aag(X1s, .(X2s))) → U4_GA(app1_in_gga(X1s, X2s))
U4_GA(app1_out_gga(Zs)) → PERM_IN_GA(Zs)
PERM_IN_GA(Xs) → U3_GA(app2_in_aag(Xs))

The TRS R consists of the following rules:

app1_in_gga(.(Xs), Ys) → U1_gga(app1_in_gga(Xs, Ys))
app1_in_gga([], Ys) → app1_out_gga(Ys)
app2_in_aag(.(Zs)) → U2_aag(app2_in_aag(Zs))
app2_in_aag(Ys) → app2_out_aag([], Ys)
U1_gga(app1_out_gga(Zs)) → app1_out_gga(.(Zs))
U2_aag(app2_out_aag(Xs, Ys)) → app2_out_aag(.(Xs), Ys)

The set Q consists of the following terms:

app1_in_gga(x0, x1)
app2_in_aag(x0)
U1_gga(x0)
U2_aag(x0)

We have to consider all (P,Q,R)-chains.

(26) MRRProof (EQUIVALENT transformation)

By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:

U3_GA(app2_out_aag(X1s, .(X2s))) → U4_GA(app1_in_gga(X1s, X2s))
U4_GA(app1_out_gga(Zs)) → PERM_IN_GA(Zs)
PERM_IN_GA(Xs) → U3_GA(app2_in_aag(Xs))

Strictly oriented rules of the TRS R:

app1_in_gga([], Ys) → app1_out_gga(Ys)
app2_in_aag(Ys) → app2_out_aag([], Ys)

Used ordering: Polynomial interpretation [POLO]:

POL(.(x1)) = 5 + x1   
POL(PERM_IN_GA(x1)) = 2 + x1   
POL(U1_gga(x1)) = 5 + x1   
POL(U2_aag(x1)) = 5 + x1   
POL(U3_GA(x1)) = x1   
POL(U4_GA(x1)) = x1   
POL([]) = 0   
POL(app1_in_gga(x1, x2)) = 4 + x1 + x2   
POL(app1_out_gga(x1)) = 3 + x1   
POL(app2_in_aag(x1)) = 1 + x1   
POL(app2_out_aag(x1, x2)) = x1 + x2   

(27) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

app1_in_gga(.(Xs), Ys) → U1_gga(app1_in_gga(Xs, Ys))
app2_in_aag(.(Zs)) → U2_aag(app2_in_aag(Zs))
U1_gga(app1_out_gga(Zs)) → app1_out_gga(.(Zs))
U2_aag(app2_out_aag(Xs, Ys)) → app2_out_aag(.(Xs), Ys)

The set Q consists of the following terms:

app1_in_gga(x0, x1)
app2_in_aag(x0)
U1_gga(x0)
U2_aag(x0)

We have to consider all (P,Q,R)-chains.

(28) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(29) TRUE

(30) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
perm_in: (b,f)
app2_in: (f,f,b)
app1_in: (b,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

perm_in_ga(Xs, .(X, Ys)) → U3_ga(Xs, X, Ys, app2_in_aag(X1s, .(X, X2s), Xs))
app2_in_aag(.(X, Xs), Ys, .(X, Zs)) → U2_aag(X, Xs, Ys, Zs, app2_in_aag(Xs, Ys, Zs))
app2_in_aag([], Ys, Ys) → app2_out_aag([], Ys, Ys)
U2_aag(X, Xs, Ys, Zs, app2_out_aag(Xs, Ys, Zs)) → app2_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_ga(Xs, X, Ys, app2_out_aag(X1s, .(X, X2s), Xs)) → U4_ga(Xs, X, Ys, X1s, X2s, app1_in_gga(X1s, X2s, Zs))
app1_in_gga(.(X, Xs), Ys, .(X, Zs)) → U1_gga(X, Xs, Ys, Zs, app1_in_gga(Xs, Ys, Zs))
app1_in_gga([], Ys, Ys) → app1_out_gga([], Ys, Ys)
U1_gga(X, Xs, Ys, Zs, app1_out_gga(Xs, Ys, Zs)) → app1_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(Xs, X, Ys, X1s, X2s, app1_out_gga(X1s, X2s, Zs)) → U5_ga(Xs, X, Ys, perm_in_ga(Zs, Ys))
perm_in_ga([], []) → perm_out_ga([], [])
U5_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) → perm_out_ga(Xs, .(X, Ys))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
app2_in_aag(x1, x2, x3)  =  app2_in_aag(x3)
.(x1, x2)  =  .(x2)
U2_aag(x1, x2, x3, x4, x5)  =  U2_aag(x4, x5)
app2_out_aag(x1, x2, x3)  =  app2_out_aag(x1, x2, x3)
U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x1, x6)
app1_in_gga(x1, x2, x3)  =  app1_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x2, x3, x5)
[]  =  []
app1_out_gga(x1, x2, x3)  =  app1_out_gga(x1, x2, x3)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x4)
perm_out_ga(x1, x2)  =  perm_out_ga(x1, x2)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(31) Obligation:

Pi-finite rewrite system:
The TRS R consists of the following rules:

perm_in_ga(Xs, .(X, Ys)) → U3_ga(Xs, X, Ys, app2_in_aag(X1s, .(X, X2s), Xs))
app2_in_aag(.(X, Xs), Ys, .(X, Zs)) → U2_aag(X, Xs, Ys, Zs, app2_in_aag(Xs, Ys, Zs))
app2_in_aag([], Ys, Ys) → app2_out_aag([], Ys, Ys)
U2_aag(X, Xs, Ys, Zs, app2_out_aag(Xs, Ys, Zs)) → app2_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_ga(Xs, X, Ys, app2_out_aag(X1s, .(X, X2s), Xs)) → U4_ga(Xs, X, Ys, X1s, X2s, app1_in_gga(X1s, X2s, Zs))
app1_in_gga(.(X, Xs), Ys, .(X, Zs)) → U1_gga(X, Xs, Ys, Zs, app1_in_gga(Xs, Ys, Zs))
app1_in_gga([], Ys, Ys) → app1_out_gga([], Ys, Ys)
U1_gga(X, Xs, Ys, Zs, app1_out_gga(Xs, Ys, Zs)) → app1_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(Xs, X, Ys, X1s, X2s, app1_out_gga(X1s, X2s, Zs)) → U5_ga(Xs, X, Ys, perm_in_ga(Zs, Ys))
perm_in_ga([], []) → perm_out_ga([], [])
U5_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) → perm_out_ga(Xs, .(X, Ys))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
app2_in_aag(x1, x2, x3)  =  app2_in_aag(x3)
.(x1, x2)  =  .(x2)
U2_aag(x1, x2, x3, x4, x5)  =  U2_aag(x4, x5)
app2_out_aag(x1, x2, x3)  =  app2_out_aag(x1, x2, x3)
U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x1, x6)
app1_in_gga(x1, x2, x3)  =  app1_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x2, x3, x5)
[]  =  []
app1_out_gga(x1, x2, x3)  =  app1_out_gga(x1, x2, x3)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x4)
perm_out_ga(x1, x2)  =  perm_out_ga(x1, x2)

(32) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

PERM_IN_GA(Xs, .(X, Ys)) → U3_GA(Xs, X, Ys, app2_in_aag(X1s, .(X, X2s), Xs))
PERM_IN_GA(Xs, .(X, Ys)) → APP2_IN_AAG(X1s, .(X, X2s), Xs)
APP2_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → U2_AAG(X, Xs, Ys, Zs, app2_in_aag(Xs, Ys, Zs))
APP2_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → APP2_IN_AAG(Xs, Ys, Zs)
U3_GA(Xs, X, Ys, app2_out_aag(X1s, .(X, X2s), Xs)) → U4_GA(Xs, X, Ys, X1s, X2s, app1_in_gga(X1s, X2s, Zs))
U3_GA(Xs, X, Ys, app2_out_aag(X1s, .(X, X2s), Xs)) → APP1_IN_GGA(X1s, X2s, Zs)
APP1_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → U1_GGA(X, Xs, Ys, Zs, app1_in_gga(Xs, Ys, Zs))
APP1_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APP1_IN_GGA(Xs, Ys, Zs)
U4_GA(Xs, X, Ys, X1s, X2s, app1_out_gga(X1s, X2s, Zs)) → U5_GA(Xs, X, Ys, perm_in_ga(Zs, Ys))
U4_GA(Xs, X, Ys, X1s, X2s, app1_out_gga(X1s, X2s, Zs)) → PERM_IN_GA(Zs, Ys)

The TRS R consists of the following rules:

perm_in_ga(Xs, .(X, Ys)) → U3_ga(Xs, X, Ys, app2_in_aag(X1s, .(X, X2s), Xs))
app2_in_aag(.(X, Xs), Ys, .(X, Zs)) → U2_aag(X, Xs, Ys, Zs, app2_in_aag(Xs, Ys, Zs))
app2_in_aag([], Ys, Ys) → app2_out_aag([], Ys, Ys)
U2_aag(X, Xs, Ys, Zs, app2_out_aag(Xs, Ys, Zs)) → app2_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_ga(Xs, X, Ys, app2_out_aag(X1s, .(X, X2s), Xs)) → U4_ga(Xs, X, Ys, X1s, X2s, app1_in_gga(X1s, X2s, Zs))
app1_in_gga(.(X, Xs), Ys, .(X, Zs)) → U1_gga(X, Xs, Ys, Zs, app1_in_gga(Xs, Ys, Zs))
app1_in_gga([], Ys, Ys) → app1_out_gga([], Ys, Ys)
U1_gga(X, Xs, Ys, Zs, app1_out_gga(Xs, Ys, Zs)) → app1_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(Xs, X, Ys, X1s, X2s, app1_out_gga(X1s, X2s, Zs)) → U5_ga(Xs, X, Ys, perm_in_ga(Zs, Ys))
perm_in_ga([], []) → perm_out_ga([], [])
U5_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) → perm_out_ga(Xs, .(X, Ys))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
app2_in_aag(x1, x2, x3)  =  app2_in_aag(x3)
.(x1, x2)  =  .(x2)
U2_aag(x1, x2, x3, x4, x5)  =  U2_aag(x4, x5)
app2_out_aag(x1, x2, x3)  =  app2_out_aag(x1, x2, x3)
U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x1, x6)
app1_in_gga(x1, x2, x3)  =  app1_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x2, x3, x5)
[]  =  []
app1_out_gga(x1, x2, x3)  =  app1_out_gga(x1, x2, x3)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x4)
perm_out_ga(x1, x2)  =  perm_out_ga(x1, x2)
PERM_IN_GA(x1, x2)  =  PERM_IN_GA(x1)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x1, x4)
APP2_IN_AAG(x1, x2, x3)  =  APP2_IN_AAG(x3)
U2_AAG(x1, x2, x3, x4, x5)  =  U2_AAG(x4, x5)
U4_GA(x1, x2, x3, x4, x5, x6)  =  U4_GA(x1, x6)
APP1_IN_GGA(x1, x2, x3)  =  APP1_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5)  =  U1_GGA(x2, x3, x5)
U5_GA(x1, x2, x3, x4)  =  U5_GA(x1, x4)

We have to consider all (P,R,Pi)-chains

(33) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

PERM_IN_GA(Xs, .(X, Ys)) → U3_GA(Xs, X, Ys, app2_in_aag(X1s, .(X, X2s), Xs))
PERM_IN_GA(Xs, .(X, Ys)) → APP2_IN_AAG(X1s, .(X, X2s), Xs)
APP2_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → U2_AAG(X, Xs, Ys, Zs, app2_in_aag(Xs, Ys, Zs))
APP2_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → APP2_IN_AAG(Xs, Ys, Zs)
U3_GA(Xs, X, Ys, app2_out_aag(X1s, .(X, X2s), Xs)) → U4_GA(Xs, X, Ys, X1s, X2s, app1_in_gga(X1s, X2s, Zs))
U3_GA(Xs, X, Ys, app2_out_aag(X1s, .(X, X2s), Xs)) → APP1_IN_GGA(X1s, X2s, Zs)
APP1_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → U1_GGA(X, Xs, Ys, Zs, app1_in_gga(Xs, Ys, Zs))
APP1_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APP1_IN_GGA(Xs, Ys, Zs)
U4_GA(Xs, X, Ys, X1s, X2s, app1_out_gga(X1s, X2s, Zs)) → U5_GA(Xs, X, Ys, perm_in_ga(Zs, Ys))
U4_GA(Xs, X, Ys, X1s, X2s, app1_out_gga(X1s, X2s, Zs)) → PERM_IN_GA(Zs, Ys)

The TRS R consists of the following rules:

perm_in_ga(Xs, .(X, Ys)) → U3_ga(Xs, X, Ys, app2_in_aag(X1s, .(X, X2s), Xs))
app2_in_aag(.(X, Xs), Ys, .(X, Zs)) → U2_aag(X, Xs, Ys, Zs, app2_in_aag(Xs, Ys, Zs))
app2_in_aag([], Ys, Ys) → app2_out_aag([], Ys, Ys)
U2_aag(X, Xs, Ys, Zs, app2_out_aag(Xs, Ys, Zs)) → app2_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_ga(Xs, X, Ys, app2_out_aag(X1s, .(X, X2s), Xs)) → U4_ga(Xs, X, Ys, X1s, X2s, app1_in_gga(X1s, X2s, Zs))
app1_in_gga(.(X, Xs), Ys, .(X, Zs)) → U1_gga(X, Xs, Ys, Zs, app1_in_gga(Xs, Ys, Zs))
app1_in_gga([], Ys, Ys) → app1_out_gga([], Ys, Ys)
U1_gga(X, Xs, Ys, Zs, app1_out_gga(Xs, Ys, Zs)) → app1_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(Xs, X, Ys, X1s, X2s, app1_out_gga(X1s, X2s, Zs)) → U5_ga(Xs, X, Ys, perm_in_ga(Zs, Ys))
perm_in_ga([], []) → perm_out_ga([], [])
U5_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) → perm_out_ga(Xs, .(X, Ys))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
app2_in_aag(x1, x2, x3)  =  app2_in_aag(x3)
.(x1, x2)  =  .(x2)
U2_aag(x1, x2, x3, x4, x5)  =  U2_aag(x4, x5)
app2_out_aag(x1, x2, x3)  =  app2_out_aag(x1, x2, x3)
U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x1, x6)
app1_in_gga(x1, x2, x3)  =  app1_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x2, x3, x5)
[]  =  []
app1_out_gga(x1, x2, x3)  =  app1_out_gga(x1, x2, x3)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x4)
perm_out_ga(x1, x2)  =  perm_out_ga(x1, x2)
PERM_IN_GA(x1, x2)  =  PERM_IN_GA(x1)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x1, x4)
APP2_IN_AAG(x1, x2, x3)  =  APP2_IN_AAG(x3)
U2_AAG(x1, x2, x3, x4, x5)  =  U2_AAG(x4, x5)
U4_GA(x1, x2, x3, x4, x5, x6)  =  U4_GA(x1, x6)
APP1_IN_GGA(x1, x2, x3)  =  APP1_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5)  =  U1_GGA(x2, x3, x5)
U5_GA(x1, x2, x3, x4)  =  U5_GA(x1, x4)

We have to consider all (P,R,Pi)-chains

(34) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 5 less nodes.

(35) Complex Obligation (AND)

(36) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

APP1_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APP1_IN_GGA(Xs, Ys, Zs)

The TRS R consists of the following rules:

perm_in_ga(Xs, .(X, Ys)) → U3_ga(Xs, X, Ys, app2_in_aag(X1s, .(X, X2s), Xs))
app2_in_aag(.(X, Xs), Ys, .(X, Zs)) → U2_aag(X, Xs, Ys, Zs, app2_in_aag(Xs, Ys, Zs))
app2_in_aag([], Ys, Ys) → app2_out_aag([], Ys, Ys)
U2_aag(X, Xs, Ys, Zs, app2_out_aag(Xs, Ys, Zs)) → app2_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_ga(Xs, X, Ys, app2_out_aag(X1s, .(X, X2s), Xs)) → U4_ga(Xs, X, Ys, X1s, X2s, app1_in_gga(X1s, X2s, Zs))
app1_in_gga(.(X, Xs), Ys, .(X, Zs)) → U1_gga(X, Xs, Ys, Zs, app1_in_gga(Xs, Ys, Zs))
app1_in_gga([], Ys, Ys) → app1_out_gga([], Ys, Ys)
U1_gga(X, Xs, Ys, Zs, app1_out_gga(Xs, Ys, Zs)) → app1_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(Xs, X, Ys, X1s, X2s, app1_out_gga(X1s, X2s, Zs)) → U5_ga(Xs, X, Ys, perm_in_ga(Zs, Ys))
perm_in_ga([], []) → perm_out_ga([], [])
U5_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) → perm_out_ga(Xs, .(X, Ys))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
app2_in_aag(x1, x2, x3)  =  app2_in_aag(x3)
.(x1, x2)  =  .(x2)
U2_aag(x1, x2, x3, x4, x5)  =  U2_aag(x4, x5)
app2_out_aag(x1, x2, x3)  =  app2_out_aag(x1, x2, x3)
U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x1, x6)
app1_in_gga(x1, x2, x3)  =  app1_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x2, x3, x5)
[]  =  []
app1_out_gga(x1, x2, x3)  =  app1_out_gga(x1, x2, x3)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x4)
perm_out_ga(x1, x2)  =  perm_out_ga(x1, x2)
APP1_IN_GGA(x1, x2, x3)  =  APP1_IN_GGA(x1, x2)

We have to consider all (P,R,Pi)-chains

(37) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(38) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

APP1_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APP1_IN_GGA(Xs, Ys, Zs)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
APP1_IN_GGA(x1, x2, x3)  =  APP1_IN_GGA(x1, x2)

We have to consider all (P,R,Pi)-chains

(39) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(40) Obligation:

Q DP problem:
The TRS P consists of the following rules:

APP1_IN_GGA(.(Xs), Ys) → APP1_IN_GGA(Xs, Ys)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(41) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

  • APP1_IN_GGA(.(Xs), Ys) → APP1_IN_GGA(Xs, Ys)
    The graph contains the following edges 1 > 1, 2 >= 2

(42) TRUE

(43) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

APP2_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → APP2_IN_AAG(Xs, Ys, Zs)

The TRS R consists of the following rules:

perm_in_ga(Xs, .(X, Ys)) → U3_ga(Xs, X, Ys, app2_in_aag(X1s, .(X, X2s), Xs))
app2_in_aag(.(X, Xs), Ys, .(X, Zs)) → U2_aag(X, Xs, Ys, Zs, app2_in_aag(Xs, Ys, Zs))
app2_in_aag([], Ys, Ys) → app2_out_aag([], Ys, Ys)
U2_aag(X, Xs, Ys, Zs, app2_out_aag(Xs, Ys, Zs)) → app2_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_ga(Xs, X, Ys, app2_out_aag(X1s, .(X, X2s), Xs)) → U4_ga(Xs, X, Ys, X1s, X2s, app1_in_gga(X1s, X2s, Zs))
app1_in_gga(.(X, Xs), Ys, .(X, Zs)) → U1_gga(X, Xs, Ys, Zs, app1_in_gga(Xs, Ys, Zs))
app1_in_gga([], Ys, Ys) → app1_out_gga([], Ys, Ys)
U1_gga(X, Xs, Ys, Zs, app1_out_gga(Xs, Ys, Zs)) → app1_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(Xs, X, Ys, X1s, X2s, app1_out_gga(X1s, X2s, Zs)) → U5_ga(Xs, X, Ys, perm_in_ga(Zs, Ys))
perm_in_ga([], []) → perm_out_ga([], [])
U5_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) → perm_out_ga(Xs, .(X, Ys))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
app2_in_aag(x1, x2, x3)  =  app2_in_aag(x3)
.(x1, x2)  =  .(x2)
U2_aag(x1, x2, x3, x4, x5)  =  U2_aag(x4, x5)
app2_out_aag(x1, x2, x3)  =  app2_out_aag(x1, x2, x3)
U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x1, x6)
app1_in_gga(x1, x2, x3)  =  app1_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x2, x3, x5)
[]  =  []
app1_out_gga(x1, x2, x3)  =  app1_out_gga(x1, x2, x3)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x4)
perm_out_ga(x1, x2)  =  perm_out_ga(x1, x2)
APP2_IN_AAG(x1, x2, x3)  =  APP2_IN_AAG(x3)

We have to consider all (P,R,Pi)-chains

(44) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(45) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

APP2_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → APP2_IN_AAG(Xs, Ys, Zs)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
APP2_IN_AAG(x1, x2, x3)  =  APP2_IN_AAG(x3)

We have to consider all (P,R,Pi)-chains

(46) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(47) Obligation:

Q DP problem:
The TRS P consists of the following rules:

APP2_IN_AAG(.(Zs)) → APP2_IN_AAG(Zs)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(48) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

  • APP2_IN_AAG(.(Zs)) → APP2_IN_AAG(Zs)
    The graph contains the following edges 1 > 1

(49) TRUE

(50) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

U3_GA(Xs, X, Ys, app2_out_aag(X1s, .(X, X2s), Xs)) → U4_GA(Xs, X, Ys, X1s, X2s, app1_in_gga(X1s, X2s, Zs))
U4_GA(Xs, X, Ys, X1s, X2s, app1_out_gga(X1s, X2s, Zs)) → PERM_IN_GA(Zs, Ys)
PERM_IN_GA(Xs, .(X, Ys)) → U3_GA(Xs, X, Ys, app2_in_aag(X1s, .(X, X2s), Xs))

The TRS R consists of the following rules:

perm_in_ga(Xs, .(X, Ys)) → U3_ga(Xs, X, Ys, app2_in_aag(X1s, .(X, X2s), Xs))
app2_in_aag(.(X, Xs), Ys, .(X, Zs)) → U2_aag(X, Xs, Ys, Zs, app2_in_aag(Xs, Ys, Zs))
app2_in_aag([], Ys, Ys) → app2_out_aag([], Ys, Ys)
U2_aag(X, Xs, Ys, Zs, app2_out_aag(Xs, Ys, Zs)) → app2_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_ga(Xs, X, Ys, app2_out_aag(X1s, .(X, X2s), Xs)) → U4_ga(Xs, X, Ys, X1s, X2s, app1_in_gga(X1s, X2s, Zs))
app1_in_gga(.(X, Xs), Ys, .(X, Zs)) → U1_gga(X, Xs, Ys, Zs, app1_in_gga(Xs, Ys, Zs))
app1_in_gga([], Ys, Ys) → app1_out_gga([], Ys, Ys)
U1_gga(X, Xs, Ys, Zs, app1_out_gga(Xs, Ys, Zs)) → app1_out_gga(.(X, Xs), Ys, .(X, Zs))
U4_ga(Xs, X, Ys, X1s, X2s, app1_out_gga(X1s, X2s, Zs)) → U5_ga(Xs, X, Ys, perm_in_ga(Zs, Ys))
perm_in_ga([], []) → perm_out_ga([], [])
U5_ga(Xs, X, Ys, perm_out_ga(Zs, Ys)) → perm_out_ga(Xs, .(X, Ys))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
app2_in_aag(x1, x2, x3)  =  app2_in_aag(x3)
.(x1, x2)  =  .(x2)
U2_aag(x1, x2, x3, x4, x5)  =  U2_aag(x4, x5)
app2_out_aag(x1, x2, x3)  =  app2_out_aag(x1, x2, x3)
U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x1, x6)
app1_in_gga(x1, x2, x3)  =  app1_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x2, x3, x5)
[]  =  []
app1_out_gga(x1, x2, x3)  =  app1_out_gga(x1, x2, x3)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x4)
perm_out_ga(x1, x2)  =  perm_out_ga(x1, x2)
PERM_IN_GA(x1, x2)  =  PERM_IN_GA(x1)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x1, x4)
U4_GA(x1, x2, x3, x4, x5, x6)  =  U4_GA(x1, x6)

We have to consider all (P,R,Pi)-chains

(51) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(52) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

U3_GA(Xs, X, Ys, app2_out_aag(X1s, .(X, X2s), Xs)) → U4_GA(Xs, X, Ys, X1s, X2s, app1_in_gga(X1s, X2s, Zs))
U4_GA(Xs, X, Ys, X1s, X2s, app1_out_gga(X1s, X2s, Zs)) → PERM_IN_GA(Zs, Ys)
PERM_IN_GA(Xs, .(X, Ys)) → U3_GA(Xs, X, Ys, app2_in_aag(X1s, .(X, X2s), Xs))

The TRS R consists of the following rules:

app1_in_gga(.(X, Xs), Ys, .(X, Zs)) → U1_gga(X, Xs, Ys, Zs, app1_in_gga(Xs, Ys, Zs))
app1_in_gga([], Ys, Ys) → app1_out_gga([], Ys, Ys)
app2_in_aag(.(X, Xs), Ys, .(X, Zs)) → U2_aag(X, Xs, Ys, Zs, app2_in_aag(Xs, Ys, Zs))
app2_in_aag([], Ys, Ys) → app2_out_aag([], Ys, Ys)
U1_gga(X, Xs, Ys, Zs, app1_out_gga(Xs, Ys, Zs)) → app1_out_gga(.(X, Xs), Ys, .(X, Zs))
U2_aag(X, Xs, Ys, Zs, app2_out_aag(Xs, Ys, Zs)) → app2_out_aag(.(X, Xs), Ys, .(X, Zs))

The argument filtering Pi contains the following mapping:
app2_in_aag(x1, x2, x3)  =  app2_in_aag(x3)
.(x1, x2)  =  .(x2)
U2_aag(x1, x2, x3, x4, x5)  =  U2_aag(x4, x5)
app2_out_aag(x1, x2, x3)  =  app2_out_aag(x1, x2, x3)
app1_in_gga(x1, x2, x3)  =  app1_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x2, x3, x5)
[]  =  []
app1_out_gga(x1, x2, x3)  =  app1_out_gga(x1, x2, x3)
PERM_IN_GA(x1, x2)  =  PERM_IN_GA(x1)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x1, x4)
U4_GA(x1, x2, x3, x4, x5, x6)  =  U4_GA(x1, x6)

We have to consider all (P,R,Pi)-chains

(53) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(54) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U3_GA(Xs, app2_out_aag(X1s, .(X2s), Xs)) → U4_GA(Xs, app1_in_gga(X1s, X2s))
U4_GA(Xs, app1_out_gga(X1s, X2s, Zs)) → PERM_IN_GA(Zs)
PERM_IN_GA(Xs) → U3_GA(Xs, app2_in_aag(Xs))

The TRS R consists of the following rules:

app1_in_gga(.(Xs), Ys) → U1_gga(Xs, Ys, app1_in_gga(Xs, Ys))
app1_in_gga([], Ys) → app1_out_gga([], Ys, Ys)
app2_in_aag(.(Zs)) → U2_aag(Zs, app2_in_aag(Zs))
app2_in_aag(Ys) → app2_out_aag([], Ys, Ys)
U1_gga(Xs, Ys, app1_out_gga(Xs, Ys, Zs)) → app1_out_gga(.(Xs), Ys, .(Zs))
U2_aag(Zs, app2_out_aag(Xs, Ys, Zs)) → app2_out_aag(.(Xs), Ys, .(Zs))

The set Q consists of the following terms:

app1_in_gga(x0, x1)
app2_in_aag(x0)
U1_gga(x0, x1, x2)
U2_aag(x0, x1)

We have to consider all (P,Q,R)-chains.